算子AB和BA的Drazin可逆性

Drazin Invertibility of Operators AB and BA

中文关键词:  算子的Drazin可逆性  算子的Drazin指标  算子的Aluthge变换.

英文关键词:Drazin invertibility of operators  Drazin index  Aluthge transforms of operators.

中文摘要:

      给定Hilbert空间${\cal H}$上的有界线性算子$A$和$B$, 本文证明了$AB$和$BA$的Drazin可逆性是等价的. 作为应用, 我们证明了$\sigma_D(AB)=\sigma_D(BA)$和$\sigma_D(A)=\sigma_D(\widetilde{A})$,这里$\sigma_D(M)$和$\widetilde{M}$分别表示算子$M$的Drazin谱和Aluthge变换.

英文摘要:

      In this note an alternative proof of the equivalence of Drazin invertibility of operators $AB$ and $BA$ is given. As an application, we will prove that $\sigma_D(AB)=\sigma_D(BA)$ and $\sigma_D(A)=\sigma_D(\widetilde{A})$, where $\sigma_D(M)$ and $\widetilde{M}$ denote the Drazin spectrum and the Aluthge transform of an operator $M\in{\cal B}({\cal H})$, respectively.